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Full text of "Analytical Mechanics"

318
ANALYTICAL MECHANICS
Making these substitutions in the torque equation we obtain
r-^
1 dt ~
mgD2
o-0           
2 sin 0 cos 9
47/r-;........;#
-   Z2-sm-2
sin (9
In actual experiments y is made less than ~     Therefore
6                                                                         *~> \i
even if the maximum value of 6 is made as large as half a radian the second term under the radical is less than , (.l0-rt- and consequently negligible. Thus the last equation reduces to
which is the well-known pendulum equation.    Therefore we have                     _ _
P-2V ~n
JL    ~~~ "
for a second approximation to the actual value of the period.
246. Torsional Pendulum.  A torsional pendulum consists of a rigid body suspended by a wire, the wire being rigidly connected to both the support and the body, Fig. 144. When the body is given an angular displacement about the wire as an axis and then left to itself it vibrates with a constant period. The torque which produces the angular displacement obeys Hooke's law; therefore